WASHINGTON, D.C. — The call that changed the course of Brett Connolly’s career came in the middle of the night.

It was around 1:30 a.m. on trade deadline day, March 2, 2015, and Connolly was still up. The Lightning returned home a few hours earlier from a Sunday road game against the Panthers in Sunrise. With the team off Monday, a bunch of players and their wives were out late together at a downtown Tampa restaurant.

Connolly was hanging out with a few friends who were in town when Lightning GM Steve Yzerman called him. Connolly had been traded to the Bruins for a couple second-round picks, which Lightning wanted to recoup after dealing a first and a third to the Flyers that same night for defenseman Braydon Coburn.

“Definitely a shock,” Connolly said.

Connolly, whom the Lightning had taken No. 6 overall in the 2010 NHL draft, was bummed. The Lightning would go on a remarkable run to the Stanley Cup final, with Coburn scoring...